Experimental Confirmation of Massive Dirac Fermions in Weak Charge-Ordering State in \alpha-(BEDT-TTF)_2I_3

The electronic structure of weak charge-ordering (CO) state just below the critical pressure in an organic conductor \alpha-(BEDT-TTF)_2I_3 was experimentally investigated using peak structure in the temperature dependence of interlayer magnetoresistance (MR). Based on a minimal model considering multiple Landau levels (LLs), we discuss herein the MR peak as characteristic to multilayer massless/massive Dirac fermion (DF) systems. MR measured in the weak CO state exhibited a clear MR peak, and its magnetic-field dependence was consistent with the LL behavior of a massive DF with a small gap. Results indicate that the weak CO state in \alpha-(BEDT-TTF)_2I_3 is a massive DF state.


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Layered organic conductor α-(BEDT-TTF)2I3 has attracted considerable attention because a two-dimensional (2D) massless Dirac fermion (DF) state occurs at high pressures [1]. At ambient pressure, it undergoes a first-order metal-insulator transition into an insulating phase caused by charge ordering (CO) at TCO = 135 K. As schematically illustrated in Fig. 1, the CO phase is suppressed by pressure, and then vanishes at approximately Pc = 1.3 GPa. It was theoretically pointed out that the metallic state above Pc is a massless DF state [2]. After that, it has been experimentally confirmed by magnetoresistance (MR) [3,4], interlayer Hall effect [5,6], specific heat [7], and NMR [8] measurements. In the massless DF state, the band dispersion around the Fermi level EF is regarded as a pair of tilted and anisotropic Dirac cones located at general points in the k space.
The interlayer MR in the massless DF phase shows characteristic features under a vertical magnetic field. It shows positive MR at low magnetic fields, which subsequently becomes negative, forming a maximum peak [3]. The negative MR is owing to the increase in degeneracy of the n = 0 Landau level (LL) in the massless DF at the quantum limit, where only n = 0 LL is partially occupied [4]. Conversely, the positive MR at low magnetic fields is because of the LL mixing owing to the non-vertical (tilting) interlayer transfer [9]. In this case, the interlayer tunneling probability depends on the vertical field, resulting in positive MR. The MR peak in the field dependence corresponds to the crossover from positive MR to negative MR.
In this study, we focused on "weak CO" state just below Pc, where the CO transition temperature TCO is largely suppressed, and the specific latent heat seems small like a second-order transition. In the weak CO state, we can expect a quasi-particle with 2D massive DF nature, that is, the dispersion where a small CO gap opens up at the Dirac 3 points of two Dirac cones (valleys), as shown in the inset of Fig. 1. The angle-dependent interlayer MR under in-plane fields, which shows continuous change between the weak CO and massless DF states [10], supports this expectation. Because the CO gap breaks the inversion symmetry, there are two types of CO domains. In the weak CO state, excess metallic conduction is often observed despite a finite gap [10,11]. This may originate from conduction along domain boundaries, which has been theoretically discussed based on the massive DF picture [12,13]. In the massive DF state, various topological properties are expected owing to their finite Berry curvature. The valley Hall effect in the trivial insulator phase [12] and the anomalous Hall effect in the Chern insulator phase [14] were theoretically discussed. Furthermore, the nonlinear anomalous Hall effect has been investigated in the current-carrying state in the weak CO state [15].
However, the massive DF in the weak CO state has not been experimentally confirmed thus far. Therefore, for confirmation, we employed the indirect method, which Sugawara et al. originally used to confirm the massless DF in -(BEDT-TTF)2I3 above Pc [16]. They noticed the peak structure in the temperature dependence of the interlayer MR, and demonstrated that the field dependence of the peak temperature, Tmax, reflects the LL of the massless DF. In this paper, we first provide the theoretical basis of the the method by Sugawara et al. [16], and extend it to a massive case. We provide a minimal model for the MR peak in temperature dependence, and show that the peak structure is characteristic to multilayer DF systems. Thereafter, we show the experimental results on the interlayer MR in the weak CO state in α-(BEDT-TTF)2I3, and experimentally confirm the massive DF using the method by Sugawara et al.
First, we present a minimal model which can reproduce the peak structure in the temperature dependence of interlayer MR in multilayer DF systems. In this model, we consider multiple LLs, but ignore the temperature dependence of scattering (mobility), the scattering broadening of LLs, and the tilting effect of the non-vertical interlayer transfer which causes interlayer tunneling between different LLs. We employ the following model of the multilayer system, where 2D massive DF layers stack with weak vertical interlayer transfer.
where ( , , ) , the matrix element of the interlayer coupling, H ⊥ , between the LL states on different layers becomes non-zero only when the 2D quantum numbers of the two states are equal and their layers are neighboring [6,9]. Simply, no LL mixing occurs 5 during interlayer tunneling in the case of vertical transfer [9]. The lowest order contribution of c t to the interlayer conductivity zz  under the vertical magnetic field is obtained by using the tunneling picture for interlayer transport as follows [17].
where  is the constant in-plane scattering time and is the Fermi distribution function. Here, we assumed that the thermal distribution width B kT is significantly larger than the LL broadening owing to in-plane scattering This formula achieves a remarkable temperature dependence characteristic of multilayer DF systems. As illustrated in Fig. 2(b), the interlayer MR exhibits a peak structure indicated by an arrow whose position depends on the magnetic field. At low temperatures, the MR exhibits insulating behavior owing to thermal activation onto the CO gap. The insulating behavior disappears in the massless DF case ( 0 = ).
The temperature dependence of the interlayer MR is caused by the thermal broadening of (2). The chemical potential  is constantly located at zero energy because of the electron-hole symmetry. The energy , which we refer to as the active region below, linearly expands with temperature, and it is represented by In fact, in a multilayer system consisting of non-Dirac semiconducting layers, we can demonstrate that the interlayer MR is almost temperature independent with no peak except for the insulating behavior. Therefore, the MR peak can be used for experimental confirmation of 2D massless/massive DFs in multilayer systems.
The peak temperatures (Tmax) obtained from Fig. 2(b) are also plotted in Fig. 2(a).  [16]. Therefore, the theoretical basis of this method has been provided, and also its general applicability to massless/massive DF systems has been confirmed.
Here, we note that the MR peak in the temperature dependence is a more general phenomenon than the peak between positive and negative MR regions in the field dependence, because the former can be reproduced without any non-vertical transfer, but the latter cannot. The present model, which considers only a vertical interlayer transfer, cannot derive the low-field positive MR and MR peak in the field dependence. Therefore, the MR peak in the temperature dependence is not necessarily the same as the MR peak in the field dependence thus, contradicting Sugawara's original idea [16].
Using Sugawara's method, we experimentally investigated whether the weak CO  from the spin polarizing of the n=0 LL were well reproduced. Although the MR in the weak CO state is approximately 10 times larger than that in the massless DF state at low magnetic fields, the MR in the weak CO state exhibits qualitatively similar behaviors as the massless DF state. Furthermore, we measured the dependence of MR on the magneticfield orientation, which is measured by the elevation angle , in the weak CO state (1.12 GPa), as shown in Fig. 3(b). Solid curves are the fitting curves of ( ) To confirm the massive DFs in the weak CO state experimentally, we measured the temperature dependence of the interlayer resistance () zz RB . The inset of Fig. 4(a) shows the temperature dependence of the interlayer resistance () zz RB at several vertical magnetic fields in the weak CO state (P = 1.12 GPa). It is remarkable that the temperature dependence of MR at each magnetic field shows a clear peak structure, whose position ( max T ) depends on the magnetic field. In addition, the MR monotonously decreases at temperatures higher than max T . These behaviors are expected in the multilayer DF system as shown in Fig. 2(b), but never in non-Dirac systems, as aforementioned.
Following Sugawara's work, we show the normalized MR   Fig. 4(a), where the effect of temperature dependence of scattering is removed. The observed result at each magnetic field is qualitatively 9 reproduced in Fig. 2(b). However, comparison between the data from different fields disagrees with Fig. 2(b). In the temperature range above max T in Fig. 4(a), the interlayer MR increases with magnetic fields simply, it shows positive MR, which is in contrast with the negative MR in Fig. 2(b). This disagreement is because the present model cannot achieve positive MR without any non-vertical transfer. Since the positive MR is caused by the 0 n  LLs and appears only in the high-temperature side of the peak, the peak temperature max T is expected not to be affected by the positive MR.
The magnetic-field dependence of the MR peak temperature max T is plotted in Fig. 4(b). The inset shows the field dependence of LLs of the massive DF with spin splitting owing to the Zeeman effect. We attempted the fitting of the following formula considering spin splitting.
Here, ( ) is the lower spin-split level of the n = 1 LL, where g = 2 is the g-factor, and B  is the Bohr magneton. The fitting curve of Eq. (3) is indicated by the solid curve labeled "massive DF" in Fig. 4(b), and it is in good agreement with the experimental data. For comparison, the fitting result assuming the massless DF (with a fixed 0 = ) is also shown by the dashed curve in Fig. 4(b), which obviously deviates from the data. as the values of the fitting parameters. The fitted value of  is rather larger than that expected from the present model, and the value of || is much smaller than the CO transition temperature TCO~25K defined by the peak of −dlogRzz/dT (Fig. 1)  The inset illustrates the 2D massive DF dispersion considered in the present model.